Mathematics for the interested outsider

Young Tableaux

We want to come up with some nice sets for our symmetric group to act on. Our first step in this direction is to define a “Young tableau”.

If is a partition of , we define a Young tableau of shape to be an array of numbers. We start with the Ferrers diagram of the partition , and we replace the dots with the numbers to in any order. Clearly, there are Young tableaux of shape if .

For example, if , the Ferrers diagram is

We see that , and so there are Young tableaux of shape . They are

We write for the entry in the place. For example, the last tableau above has , , and .

We also call a Young tableau of shape a “-tableau”, and we write . We can write a generic -tableau as .

[…] as a quick use of this concept, think about how to fill a Ferrers diagram to make a standard Young tableau. It should be clear that since is the largest entry in the tableau, it must be in the rightmost […]

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.